Question: Simplify; express your answer in exponential form. Assume $x\neq 0, z\neq 0$. $\dfrac{{(x^{-1})^{2}}}{{(x^{-2}z^{5})^{3}}}$
Explanation: To start, try working on the numerator and the denominator independently. In the numerator, we have ${x^{-1}}$ to the exponent ${2}$ . Now ${-1 \times 2 = -2}$ , so ${(x^{-1})^{2} = x^{-2}}$ In the denominator, we can use the distributive property of exponents. ${(x^{-2}z^{5})^{3} = (x^{-2})^{3}(z^{5})^{3}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(x^{-1})^{2}}}{{(x^{-2}z^{5})^{3}}} = \dfrac{{x^{-2}}}{{x^{-6}z^{15}}}$ Break up the equation by variable and simplify. $\dfrac{{x^{-2}}}{{x^{-6}z^{15}}} = \dfrac{{x^{-2}}}{{x^{-6}}} \cdot \dfrac{{1}}{{z^{15}}} = x^{{-2} - {(-6)}} \cdot z^{- {15}} = x^{4}z^{-15}$.